Linear time algorithm for quantum 2SAT

A canonical result about satisfiability theory is that the 2-SAT problem can be solved in linear time, despite the NP-hardness of the 3-SAT problem. In the quantum 2-SAT problem, we are given a family of 2-qubit projectors $\Pi_{ij}$ on a system of $n$ qubits, and the task is to decide whether the Hamiltonian $H=\sum \Pi_{ij}$ has a 0-eigenvalue, or it is larger than $1/n^\alpha$ for some $\alpha=O(1)$. The problem is not only a natural extension of the classical 2-SAT problem to the quantum case, but is also equivalent to the problem of finding the ground state of 2-local frustration-free Hamiltonians of spin $\frac{1}{2}$, a well-studied model believed to capture certain key properties in modern condensed matter physics. While Bravyi has shown that the quantum 2-SAT problem has a classical polynomial-time algorithm, the running time of his algorithm is $O(n^4)$. In this paper we give a classical algorithm with linear running time in the number of local projectors, therefore achieving the best possible complexity.Comment: 20 page

Hypoplastic amelogenesis imperfecta with multiple impacted teeth: report of two cases

Amelogenesis Imperfecta (AI) represents a group of developmental conditions, genomic in origin, which affect the structure and clinical appearance of enamel of all or nearly all the teeth in a more or less equal manner. It is usually inherited either as an X-linked, autosomal dominant or autosomal recessive trait. The enamel may be hypoplastic, hypomineralised or both and affected teeth may be discolored, sensitive or prone to disintegration. Diagnosis is based on the family history, pedigree plotting and meticulous clinical observation. The treatment of patients with AI should start with early diagnosis and intervention to prevent latter restorative problems. Herein, we present two case reports of hypoplastic amelogenesis imperfecta with oligodontia, multiple unerupted teeth, pulpal calcification, taurodontism and anterior deep bite who were provided with functional and esthetic rehabilitation

On the complexity of probabilistic trials for hidden satisfiability problems

What is the minimum amount of information and time needed to solve 2SAT? When the instance is known, it can be solved in polynomial time, but is this also possible without knowing the instance? Bei, Chen and Zhang (STOC '13) considered a model where the input is accessed by proposing possible assignments to a special oracle. This oracle, on encountering some constraint unsatisfied by the proposal, returns only the constraint index. It turns out that, in this model, even 1SAT cannot be solved in polynomial time unless P=NP. Hence, we consider a model in which the input is accessed by proposing probability distributions over assignments to the variables. The oracle then returns the index of the constraint that is most likely to be violated by this distribution. We show that the information obtained this way is sufficient to solve 1SAT in polynomial time, even when the clauses can be repeated. For 2SAT, as long as there are no repeated clauses, in polynomial time we can even learn an equivalent formula for the hidden instance and hence also solve it. Furthermore, we extend these results to the quantum regime. We show that in this setting 1QSAT can be solved in polynomial time up to constant precision, and 2QSAT can be learnt in polynomial time up to inverse polynomial precision.Comment: 24 pages, 2 figures. To appear in the 41st International Symposium on Mathematical Foundations of Computer Scienc

Orbital MALT Lymphoma: A Case Report

A case of orbital MALT (mucous associated lymphoid tissue) lymphoma is reported for its rarity. It presented as a large tumor obscuring the whole eye with loss of vision, without any signs of dissemination and remained free of recurrence or metastasis 12 months after undergoing simple surgical excision

Genetic Variation in the IL-6 and HLA-DQB1 Genes Is Associated with Spontaneous Clearance of Hepatitis C Virus Infection.

Background. Millions of people are infected with hepatitis C virus (HCV) worldwide and 30% spontaneously clear the infection. Reasons for HCV clearance without antiviral treatment are not well understood. Methods. Blood was collected for DNA analysis from patients with chronic HCV infection or evidence of spontaneous clearance. To overcome anticipated limitations of small sample size, primary analyses consisted of a candidate gene analysis of 12 preselected genes based on known association with host immunologic response to HCV infection. To further reduce the impact of multiple testing on power, a single likelihood ratio test was conducted for each gene using all associated SNPs assayed on the Illumina Quad 610/660W chip. Step-down permutation methods were used to adjust for multiple testing in all analyses. Results. Ninety-five and 62 patients with HCV chronic infection or spontaneous clearance, respectively, were included for analysis. HLA-DQB1 (p = 1.76⁎10(-5)) and IL-6 (p = 0.0007) genes were significantly associated with spontaneous HCV clearance. IL-28B was not significantly associated with spontaneous clearance (p = 0.17). Conclusion. Our whole-gene analytic strategy identified a previously unreported association of IL-6 with spontaneous clearance of HCV infection. We also confirmed the finding that HLA-DQB1 is associated with spontaneous resolution of HCV infection